The given equation is:

$3(x + 9)^{\frac{3}{4}} = 24$

Let's solve this step by step.

Step 1: Divide both sides of the equation by 3 to simplify:

$(x + 9)^{\frac{3}{4}} = \frac{24}{3}$ $(x + 9)^{\frac{3}{4}} = 8$

Step 2: Raise both sides to the power of $\frac{4}{3}$ to eliminate the exponent:

$(x + 9) = 8^{\frac{4}{3}}$

Step 3: Calculate $8^{\frac{4}{3}}$:

$8^{\frac{4}{3}} = (2^3)^{\frac{4}{3}} = 2^4 = 16$ So, $x + 9 = 16$

Step 4: Solve for $x$:

$x = 16 - 9$ $x = 7$

Thus, the solution is $x = 7$. The correct answer is C. 7.

Would you like further details on any part of this solution?

Here are some related questions to deepen your understanding:

1. How do you solve an equation with fractional exponents in general?
2. What is the process to simplify fractional exponents?
3. What if the exponent had been a different fraction, such as $\frac{2}{3}$?
4. Can you explain the relationship between radical expressions and fractional exponents?
5. What would happen if the equation had a negative exponent?

Tip: When dealing with fractional exponents, remember that the denominator indicates the root, and the numerator indicates the power.